# A plausible upper limit on the number of e-foldings

###### Abstract

Based solely on the arguments relating Friedmann equation and the Cardy formula we derive a bound for the number of e-folds during inflation for a standard Friedmann-Robertson-Walker as well as non-standard four dimensional cosmology induced by a Randall-Sundrum-type model.

###### pacs:

04.70.Dy, 98.80.CqMotivated by the well-known example of black hole entropy, an influential holographic principle has put forward, suggesting that microscopic degrees of freedom that build up the gravitational dynamics actually reside on the boundary of space-time thooft . This principle developed to the Maldacena’s conjecture on AdS/CFT correspondence maldacenaconj and further very important consequences, such as Witten’s wittenidentification identification of the entropy, energy and temperature of CFT at high temperatures with the entropy, mass and Hawking temperature of the AdS black hole hawpage .

On the other hand, although standard model of particle physics has been established as the uncontested theory of all interactions down to distances of m, there are good reasons to believe that there is a new physics arising soon at the experimental level, and this fact is going to appear in a clear fashion in the setup of cosmology, where a large number of quite exciting developments is giving rise to a precision cosmology. Thus cosmology may provide an alternative laboratory for string theory. From the eighties several authors tried to analyse the kind of cosmology arising from string inspired models, which are essentially general relativity in higher dimensions together with scalar and tensor fields. In case we introduce also the brane concept, a consistent picture of the brane universe is achieved, and we can describe the evolution of the universe by means of solutions of the Einstein field equations in higher dimensions, with a four dimensional membrane. We thus seek at a description of the powerful holographic principle in cosmological settings, where its testing is subtle, and the question of hologrphy therein has been considered by several authors severalauthors have shown that for flat and open Friedmann-Lemaitre-Robertson-Walker(FLRW) universes the area of the particle horizon should bound the entropy on the backward-looking light cone. In addition to the study of holography in homogeneous cosmologies, attempts to generalize the holographic principle to a generic realistic inhomogeneous cosmological setting were carried out in tavaellis . Later, the very interesting study of the holographic principle in FLRW universe filled with CFT with a dual AdS description has been done by Verlinde everlindeunpu , revealing that when a universe-sized black hole can be formed, an interesting and surprising correspondence appears between entropy of CFT and Friedmann equation governing the radiation dominated closed FLRW universes. Generalizing Verlinde’s discussion to a broader class of universes including a cosmological constant wangabdallasu , we found that matching of Friedmann equation to Cardy formula holds for de Sitter closed and AdS flat universes. However for the remaining de Sitter and AdS universes, the argument fails due to breaking down of the general philosophy of the holographic principle. In high dimensions, various other aspects of Verlinde’s proposal have also been investigated in a number of works all1 .

In a recent paper savoverli , further light on the correspondence between Friedmann equation and Cardy formula has been shed from a Randall-Sundrum type brane-world perspective randallsundrum . Considering the CFT dominated universe as a co-dimension one brane with fine-tuned tension in a background of an AdS black hole, Savonije and Verlinde found the correspondence between Friedmann equation and Cardy formula for the entropy of CFT when the brane crosses the black hole horizon. This result has been further confirmed by studying a brane-universe filled with radiation and stiff-matter, quantum-induced brane worlds and radially infalling brane biswasmukh . The discovered relation between Friedmann equation and Cardy formula for the entropy shed significant light on the meaning of the holographic principle in a cosmological setting. However the general proof for this correspondence for all CFTs is still difficult at the moment. Other settings have also been considered as in e. g. all2 . It is worthwhile to further check the validity of the correspondence in broader classes of situations than everlindeunpu ; savoverli .

Our motivation here is the use of the correspondence between the CFT gas and the Friedmann equation and to establish an upper bound for the number of e-foldings during inflation, using a small number of assumptions. The main point is an upper limit for entropy, a fact that we can derive from the above correspondence. Recently, Banks and Fischler banksfischler have considered the problem of the number of e-foldings in a universe displaying an asymptotic de Sitter phase, as our own. As a result the number of e-foldings is not larger than 65/85 depending on the type of matter considered.

Here we reconsider the problem from the point of view of the entropy content of the Universe, and considering the correspondence between the Friedmann equation and Cardy formula in Brane Universes, as discussed by us binabdsu .

The main points in our argument are the following. First we assume a FRW closed universe with a positive cosmological constant which does not recollapse, as implied by some recent observations. Such an assumption is crucial to our argument. Further on, we assume that there is an upper bound for entropy in the Universe. Such an entropy is obtained from the bulk black hole in the sense of holography and considered to have a bound on its storage to prevent the collapse of the universe. These hypothesis are sufficient conditions for us to arrive at the result. They are nevertheless not necessary, but we think them to be quite natural. Later on, we extend the result for a Randall-Sundrum brane world model. While dynamical details of the AdS/CFT correspondence have been used in the derivation, it is the bound on the entropy of the Universe which is the essential ingredient.

The existence of an upper bound for the number of e-foldings before the end of inflation was also studied recently in scott and liddle . However in their investigation the number of e-foldings is related both to a possible reduction in energy scale during the late stages of inflation and to the complete cosmological evolution, being model dependent. The bound has been obtained in some very simple cosmological settings, while it is still difficult to be obtained in nonstandard models. Using the entropy bound, the consideration of physical details connected with the universe evolution can be avoided. We have obtained the upper bound for the number of e-foldings for a standard FRW universe as well as non-standard cosmology based on the brane inspired idea of Randall and Sundrum models.

The starting point is that the scalar factor, in case of the brane cosmology, is defined by the Darmois-Israel condition darmoisisrael binabdsu eabdbertha . We consider a bulk metric defined by

(1) |

where is the curvature radius of AdS spacetime. takes the values corresponding to flat, open and closed geometrics, and is the corresponding metric on the unit three dimensional plane, hyperboloid or sphere. The black hole horizon is located at and

(2) |

The relation between the parameter and the Arnowitt-Deser-Misner (ADM) mass of the five dimensional black hole is cejm

(3) |

where is the volume of the unit 3 sphere, , and is the Newton constant in the bulk. It is related to the Newton constant on the brane as .

Here, the location and the metric on the boundary are time dependent. We can choose the brane time such that , in which case the metric on the brane is given by

(4) |

The Conformal Field Theory lives on the brane, which is the boundary of the AdS hole. The energy for a CFT on a sphere with radius , of volume is given by . The total energy is not a constant during the cosmological expansion, but decreases like . This is consistent with the fact that for a CFT energy density we have

(5) |

The entropy of the CFT on the brane is equal to the Bekenstein-Hawking entropy of the AdS black hole wittenidentification garrigasasaki , which is given by the area of the horizon measured in bulk Planckian units, as given by

(6) |

The area of a 3-sphere in AdS equals the volume of the corresponding spatial section for an observer on the brane.

The total entropy is a constant during the cosmological evolution, but the entropy density of the CFT on the brane is

(7) |

In the brane world interpretation we have to fulfill matching conditions for the gravitational fields due to the immersion of the brane into the bulk (see e.g. binabdsu ,darmoisisrael ,eabdbertha ). From the matching conditions we find now the cosmological equations in the brane are

(8) |

where is the critical brane tension. Taking , (8) reduces to the Friedmann equation of CFT radiation dominated brane universe without cosmological constant discussed in savoverli . If or , the brane-world is a de Sitter universe or AdS universe, respectively. Using (5) the Friedmann equation can be written in the form

(9) |

where is the effective positive cosmological constant in four dimensions, in agreement with observations. The arguments and formulae above depend on the holographic properties, which we suppose to be valid in the theory.

The relation between the energy density and the entropy, can be used to rewrite the Friedmann equation as

(10) |

which corresponds to the movement of a mechanical nonrelativistic particle in a given potential. This equation is crucial for the developments which follow, being deeply rooted on (7), which is a direct consequence of holography and the subsequent construction, as given in e. g. binabdsu . For a closed universe there is a critical value for which the solution extends to infinity (no big crunch) which is

(11) |

is the size of the de Sitter horizon, which is the box holding the maximum amount of the entropy. The above equation was obtained by considering the potential obtained from the mechanical problem (10) for a closed universe, namely , which is and divides the collapsing region (for smaller than the one corresponding to the maximum) or an expanding region, (for larger than the one corresponding to the maximum). For the expanding case the first inequality in (11) must be fulfilled. . We want a solution which does not collapse to zero, but rather develops to infinity. The maximum of the potential occurs at

Now, dividing (7) by (5) we get , thus the entropy in such a universe can be rewriten as

(12) |

at the end of inflation. We take to be the energy density during inflation, that is, , which for the scale factor at the exit of inflation leads to the value , where corresponds to the apparent horizon during inflation, and we obtain

(13) |

(14) |

from which we arrive at

(15) |

where we used the usual values GeV and eV. Note that the bound (14) is stricter than the previous used before. The bound (11) implies a bound for the scale factor such that only limited amount of entropy can store to avoid the big crunch. This is the reason we need (14) in order to get the result (15).

For this standard FRW universe, the bound obtained is in agreement with scott and liddle as well as with banksfischler for a universe filled with radiation. The de Sitter-closed universe satisfies the correspondence between Friedmann equation and Cardy formula, which is the extension of Verlinde’s argument (see wangabdallasu ) showing the spirit of holography. It is doubtful that a similar bound can be obtained along the same lines for open or flat universes. However, we stress the fact that recent WMAP analysis favours a closed universe, although this is still a result to be further confirmed nature .

Let us consider now very high energy brane corrections to the Friedmann equation. From the Darmois-Israel conditions we find

(16) |

where is the brane tension and in the very high energy limit the term dominates. and are four-dimensional Planck scale and cosmological constant, respectively. Within the high-energy regime, the expansion laws corresponding to matter and radiation domination are slower than in the standard cosmology liddle . Slower expansion rates lead to a larger value of the number of e-foldings. However, the full calculation has not been obtained due to the lack of knowledge of this high-energy regime. Here we study this problem from the point of view of holography.

The energy density of the CFT and the entropy density are related as follows,

(17) |

which can be substituted in the Friedmann equation as before, leading to a bound for the entropy, as well as a bound for the scale factor, as given by

(18) |

We consider the era when the quadratic energy density is important. The brane tension is required to be bounded by maartenswands . Combining the values of and we chose before, a bound for is given by

(19) |

The number of e-foldings obtained is bigger than the value in standard FRW cosmology, which is consistent with the argument of liddle .

In summary, we have derived the upper limit for the number of e-foldings based upon the arguments relating Friedmann equation and Cardy formula. For the standard FRW universe our result is in good agreement with scott and liddle and in the radiation dominated case with banksfischler . For the brane inspired cosmology in four dimensions we obtained a larger bound. Considering such a high energy context, the expansion laws are slower than in the standard cosmology, and our result can again be considered to be consistent with the argument in liddle . The interesting point here is that using the holographic point of view, we can avoid a complicated physics during the universe evolution and give a reasonable value for the upper bound of the number of e-foldings. Elsewhere scott ; liddle the mechanism of ending inflation and the reheating phase are very important. Therefore in those discussions there is a strong model dependence. In the present description using the holographic description we do not refer to those sensitive processes. We thus claim that this discussion is more general.

ACKNOWLEDGEMENT: This work was partically supported by Fundacão de Amparo à Pesquisa do Estado de São Paulo (FAPESP) and Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPQ). B. Wang would like to acknowledge the support given by NNSF, China, Ministry of Science and Technology of China under Grant No. NKBRSFG19990754 and Ministry of Education of China. We would like also to thank S. Nojiri for a useful correspondence.

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